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IV. CONCLUSIONS AND FUTURE DIRECTIONS

IV. CONCLUSIONS AND FUTURE DIRECTIONS

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Broadband Transient Infrared Spectroscopy



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ACKNOWLEDGMENTS



The author is indebted to the hard work and extreme experimental efforts

of my many colleagues without whom this research would never have

been possible. These include NIST/NRC postdoctoral research associates

Drs. Steven Arrivo, Tom Dougherty (who passed away in 1997), Tandy

Grubbs, Todd Heimer, and Andrea Markelz, guest researcher Dr. Valeria

Kleiman, and collaborators Prof. Ted Burkey, Dr. Joe Melinger, and

Dr. Michael George. I am also indebted to my esteemed colleague Dr. John

Stephenson, Group Leader of the NIST Laser Applications Group, for his

continual support of this research and invaluable scientific discussions.

Research funding for most of this work was provided by internal NIST

STRS support and the NIST Advanced Technology Program.

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4

The Molecular Mechanisms Behind

the Vibrational Population Relaxation

of Small Molecules in Liquids

Richard M. Stratt

Brown University, Providence, Rhode Island



I. INTRODUCTION



The question of how fast a vibrationally hot molecule loses its excess

energy — or, better yet, how fast any one piece of a molecule cools

down — is central to chemical reaction dynamics (1–8). Not only is the

inverse, but closely related problem of how fast vibrational energy can be

added to a molecule at the very core of what chemical reactions are all

about, it is the ability to disperse the excess thermal energy produced in a

reaction that prevents the product version of our molecules from promptly

reverting to their nascent reactant forms (9–11).

The issues raised in pursuing these problems are particularly interesting because they highlight the striking differences (and the curious similarities) between isolated-molecule dynamics and that seen in liquids. In

a gas-phase bimolecular reaction, the two colliding partners can dispose

of their excess internal energy by converting it into translational or rotational kinetic energy. But are these same options going to be open to

systems in the tightly packed confines of a dense liquid, where there is

neither free translation nor free rotation? Worse still, will the inability of

the products to escape physical contact until they diffuse apart prevent

such simple kinematic mechanisms from operating (12)? Besides, even if



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Stratt



our molecules found a way to rid themselves of their unwanted vibrational

energy, how easily could the surrounding liquid absorb it? The energy of

a single quantum of a CO stretch is 10 times what kB T is at room temperature. In effect, shedding a quantum of vibration is equivalent to plunging

a red-hot iron into our cold liquid. Should we think of the process as

instantaneously vaporizing the surrounding solvent?

The prevailing theoretical models used to confront these issues reflect

this profound dichotomy between gas-phase and condensed-phase perspectives. Historically, the isolated binary collision (IBC) picture of energy

transfer has been the most frequently invoked scheme for calculating rates in

liquids (1,13–17), yet it has all the earmarks of a quintessentially gas-phase

treatment: molecules are assumed to lose vibrational energy through discrete

(binary) collisions with individual solvent molecules. Not every such collision is likely to be equally effective, but one might surmise, as is the case

in the gas phase, that the fraction of collisions that do succeed in transferring energy might be largely independent of the features of the surrounding

solvent. If one proceeds with this assumption and blithely goes on to regard

the collisions as completely uncorrelated (18–20), the net result is that the

rate of energy loss by a dissolved vibrating molecule can be written as

the simple product of the collision rate — a condensed-phase quantity, but

one independent of any specifics of vibrational dynamics — and the fraction of effective collisions — something reflecting the details of what the

two-body dynamics would be in isolation.

Probably the most appropriate first response to this model is to regard

it as contrary to any legitimate, microscopically detailed view of what

liquids are about (21–23). The principal problem is that it is not at all

clear that there is such a thing as a well-defined collision in a dense liquid.

Each molecule in a liquid is constantly being jostled by on the order of a

dozen neighboring molecules. Indeed, were we to try to define “collisions”

strictly as changes in our solute’s potential energy caused by motion of the

solvent, we would be hard pressed to find a time when collisions were not

happening (24).

We should hasten to note that these fundamental difficulties do not

mean that this theory does not often “work.” The most common application of IBC theory points to its particularly simple prediction for the

dependence of relaxation rates on the thermodynamic state of the solvent:

with the Enskog estimate of collision rates, the ratio of vibrational relaxation rates at two different liquid densities 1 and 2 is just the ratio of the

local solvent densities [ 1 g1 R / 2 g2 R ], where g r is the solute-solvent

radial distribution function and R defines the solute-solvent distance at



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which an energy-relaxing collision is presumed to take place (13,22,25).

These kinds of ratios of rates are, in fact, often well predicted by such

expressions. There is some evidence as well that ratios of rates stemming from different excited vibrational states can also be rationalized quite

nicely (16). The fact remains though, that because of its gas-phase roots,

the IBC theory per se could never provide us with the tools necessary to

discover the genuine molecular mechanisms by which vibrational energy

relaxes in liquids. The need to postulate an arbitrary solute-solvent collision

distance R (and the extraordinary sensitivity to the precise value chosen)

serves as a warning sign that the theory does not bear too close an examination (15).

By the same token, of course, just because a model has its home in

the condensed-phase world does not mean that it is any more suited to our

purposes. It is not impossible to think about vibrational energy relaxation

from a diametrically opposite limit, to regard the relaxation as a dissipation

of heat into the surrounding, more or less continuous, medium (26,27).

Vibrational dephasing rates have actually been predicted based on the

values of a variety of different macroscopic transport coefficients of the

solvent (25). As with the IBC approach, such models will neatly circumvent the need to understand the microscopic details of dynamics in a liquid,

but for the same reasons, these continuum models are going to be fundamentally incapable of telling us which solvent molecules are doing what or

when they are doing it. To get at mechanistic questions this specific — and

even to find out how to whether it is useful to try to be this molecularly

detailed in the ever-changing environment of a liquid — we need to pursue

a more broadly based statistical mechanical approach to liquid dynamics.

This chapter is an attempt to summarize some of the recent progress that has

been made in understanding the actual mechanisms of vibrational energy

relaxation using one such approach.

We should emphasize that the work discussed in this chapter is rather

limited in scope. With few exceptions it will be concerned with obtaining

purely classical mechanical perspectives on the very simplest example of

vibrational energy relaxation — that of diatomic solutes dissolved in simple

atomic and molecular liquids. Such problems do not come close to spanning

the range of interesting topics suggested by modern infrared and Raman

spectroscopies, but they are among the first examples of solute relaxation

processes for which it has been possible to elucidate molecular mechanisms.

Besides, beginning at the beginning is not necessarily a bad approach. We

will return in the final portion of the chapter to the prospects for taking a

somewhat wider view.



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Stratt



II. VIBRATIONAL FRICTION

A. Vibrational Energy Relaxation and Vibrational Friction



A powerful way of simultaneously including molecular and macroscopic

perspectives is to write the exact equation of motion for the interesting

degree of freedom — for us, the normal coordinate of the relevant solute

vibrational mode — allowing the influence of the solvent to make an

appearance only through one of a number of different kinds of effective

forces. Generalized Langevin equations, (e.g., see Reference 28) express

the (classical) equation of motion for the special coordinate x in terms of

the potential of mean force W x , the potential energy x would feel were

the solvent equilibrated around the solute, and the residual forces resulting

explicitly from the solvent dynamics:

m d2 x/dt2 D



t



∂W/∂x



dt0 Á t



t0 v t0 C F t



(1)



0



Here m is the mass and v is the velocity dx/dt associated with the x

coordinate. The remaining, dynamically induced solvent effects show up in

F t , the so-called fluctuating force, and Á t , the dynamical friction. Indeed,

the last two terms in Equation (1) are the key — without them there would

be no mechanism for a solvent to accept energy from a solute (29,30).

One of the features that makes Equation (1) such a good starting

point for our work is that it can be, in principal, exact. It is possible to

show, without ever explicitly evaluating F and Á, that these crucial functions

really do exist and are well defined (31). These formal definitions are rarely,

if ever, useful in practical numerical calculations, but one can also work

backwards from the exact dynamics x t (e.g., from a molecular dynamics

simulation) to derive what the friction in particular must look like (32).

The analysis tells us, moreover, that the exact F and Á are actually related

to one another (28,31). The requirement that the relaxed system must be in

equilibrium at some temperature T can be shown to set the magnitude and

correlations of the fluctuating force:

hF 0 F t i D kB T Á t



(2)



with kB being Boltzmann’s constant and the brackets representing an equilibrium average. The upshot is that if we can understand the specifics of

the vibrational friction, we should be able to predict the desired vibrational

relaxation rates.

Interestingly, this microscopic friction behaves much the way our

macroscopic intuitions predict that it should. The manner in which the



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153



friction appears on the right-hand side of Equation (1) says that it leads

to an effective force proportional to the mode velocity v, but opposing

it — much as one would expect from some sort of frictional drag. The

fact that this drag has a time delay, that the drag at time t results from a

velocity at an earlier time t0 [which makes Equation (1) a generalized rather

than an ordinary Langevin equation], might seem a bit of a complication,

but it too is eminently reasonable. One can think of any motion of the

solute mode, v, as perturbing the solvent away from its preferred parts of

phase space. The solvent, in its best LeChatelier fashion, reacts to restore

the status quo by evolving in such a way as to penalize any subsequent

motion of the mode — that is, it generates a frictional drag. However, in

any genuinely molecular picture, the effects of this solvent back-reaction

cannot be instantaneous; it has to have a time lag commensurate with the

time scales on which the solvent moves. Much of our study of vibrational

relaxation can therefore be interpreted as an investigation into just what

these time scales are.

This conceptual link between the solvent vibrational friction and

vibrational energy relaxation is actually mirrored by an important practical connection. Within the rather accurate Landau-Teller approximation,

(29,33,34), the rate of vibrational energy relaxation for a diatomic with

frequency ω0 and reduced mass is given by

1

D

T1



1



ÁR ω0



(3)



where ÁR ω is the cosine transform of the vibrational friction

1



ÁR ω D



dt cos ωt Á t



(4)



0



In other words, the ability of the solvent to absorb a quantum of energy h¯ ω0

(or its classical equivalent) is determined quite literally by the ability of the

solvent to respond to the solute dynamics at a frequency ω D ω0 . One can

derive this relation quantum mechanically by assuming that the solvent’s

effect on the solute can be handled perturbatively within Fermi’s golden

rule (1), but it is actually more general than that. Perhaps it is worth pausing

to see how the same basic result appears in a purely classical context.

Quite generally we can imagine the Hamiltonian for our system as a

sum of Hu , a Hamiltonian for the solute vibration, Hv , a Hamiltonian for

the solvent, and Vc , the piece of the potential energy coupling the two:

H D Hu p, x C Hv p, q C Vc x, q



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(5)



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